In the second-moment-closure (SMC) method of turbulence modelling, measures to ensure a realizable turbulence model are currently limited to constraining the Reynolds stress to physically plausible values. These constraints address neither the realizability of the other statistical moments (e.g. pressure–strain correlation) nor the underlying causes of unrealizable Reynolds stress. For achieving increased consistency with flow physics in SMC, we propose the additional requirement that the closure model for each of the unclosed statistical moments in the Reynolds stress equation be individually realizable. We then proceed to derive two realizability constraints on the rapid-pressure statistics: (i) the rapid pressure-gradient variance must be positive which leads to the requirement that the $M_{ijkl}$ tensor must be positive semi-definite, and (ii) the rapid pressure–strain correlation closure must satisfy the Schwarz inequality. Calculations with currently popular models show that unrealizable rapid-pressure–strain correlation precedes unrealizable Reynolds stress. It is also demonstrated that when the Launder, Reece and Rodi (LRR) rapid-pressure–strain correlation model is modified (truncated) to satisfy the new constraints, Reynolds stress realizability is always preserved. These findings clearly indicate that an unrealizable closure model is the cause of Reynolds stress realizability violation and highlight the importance of the new constraints.